HFCC Math Lab Arithmetic - 8. Proportion Word Problems. 1. Represent the unknown quantity by the variable x.

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1 HFCC Math Lab Arithmetic - 8 To solve word problems using proportions: Proportion Word Problems 1. Represent the unknown quantity by the variable x. 2. Set up a ratio using the given rate. 3. Set up a ratio involving the variable x. 4. Form a proportion by setting the two ratios from 2 and 3 equal to each other. Include the units of the quantities when you write the proportion. Be sure that the same units occupy corresponding positions in the two ratios of the proportion. CORRECT The units occupy corresponding positions in the 2 ratios. INCORRECT The units DO NOT occupy corresponding positions in the 2 ratios miles miles miles hours hours hours hours miles wins wins wins games games games games wins dollars dollars dollars ounces ounces ounces ounces dollars 5. Once the quantities have been correctly entered into the proportion by using the units as a guide, drop the units. Cross-multiply to solve for x. 6. Write a sentence for the answer. Now, let s see how this procedure can be applied to solve some actual proportion word problems. Revised 05/10 1

2 Ex 1: A man used 10 gallons of gasoline on a 180-mile trip. How many gallons of gasoline will he use on a 450-mile trip? 1. Let x represent the number of gallons of gasoline to be used on the 450 mile trip. 2. Set up a ratio using the given rate. 3. Set up a ratio involving x. x gallons 450 miles 10 gallons 180 miles 10gallons x gallons 180 miles 450 miles 10 x Cross-multiply: 180 ( x ) = 10 (450) Solve for x: 180x x x On a 450-mile trip, the man will use 25 gallons of gas. Ex 2: A baseball team wins seven of its first twelve games. How many games would you expect the team to win out of its thirty-six games if the team continues to play with the same degree of success? 1. Let x represent the number of games the baseball team expects to win out of its thirty-six games. 7 wins 12 games x wins 36 games 7 wins x wins 12 games 36 games Revised 05/10 2

3 7 x Cross-multiply: 12( x ) 7(36) Solve for x: 12x x x The team can expect to win 21 out of its 36 games. Ex 3: The property tax on a $60,000 home is $1,500. At this rate what will the property tax be on a home worth $75,000? 1. Let x represent the property tax on a $75,000 home. $1,500 tax $60,000 home $ x tax $75,000 home $1,500 tax $ x tax $60, 000 home $75,000 home 1,500 x 60, ,000 Cross-multiply: 60, 000( x) 1500(75, 000) Solve for x: 60, 000x 112,500, , 000x 112,500, , , 000 x 1, The property tax on a home worth $75,000 will be $1,875. Revised 05/10 3

4 Ex 4: If a piece of steel 12 feet long weighs 168 pounds, how much will a piece of steel 20 feet long weigh? 1. Let x represent the number of pounds that the 20-foot long piece of steel weighs. 12 feet 168 pounds 20 feet x pounds 12 feet 20 feet 168 pounds x pounds x Cross-multiply 12( x ) 20(168) Solve for x: 12x 3,360 12x 3, x A piece of steel 20 feet long will weigh 280 pounds. Ex 5: How many gallons of paint will you have to purchase to cover 5,500 square feet of a surface if 2 gallons of paint will cover 700 square feet? 1. Let x represent the number of gallons of paint necessary to cover 5500 square feet. 2 gallons 700 square feet x gallons 5,500 square feet 2 gallons x gallons 700 square feet 5,500 square feet Revised 05/10 4

5 2 x 700 5,500 Cross-multiply: x (700) 2(5,500) Solve for x: 700x 11, x 11, x You will have to purchase 16 gallons of paint to cover 5,500 square feet. Ex 6: A man 6 feet tall casts a shadow 4 feet long. What is the length of a shadow cast at the same time by a statue that is 18 inches high? 1. Let x feet represent the length of the of the shadow cast by the statue. 6feet tall 4- foot shadow 18 inches tall x-foot shadow 6feet tall 4-foot shadow 18 inches tall x-foot shadow We need to change 6 feet to 72 inches so that the units are in agreement throughout 72 inches 18 inches the proportion as well as occupying corresponding positions. 4 feet x feet x Cross-multiply: x (72) 18(4) Solve for x: 72x 72 72x x 1 6. The statue casts a 1-foot shadow. Revised 05/10 5

6 Ex 7: The dosage of a certain medication is 2 ounces for every 50 pounds of body weight. How many ounces of the medication are required for a person who weighs 175 pounds? 1. Let x represent the number of ounces of the medication required for a 175-pound person. 2 ounces of medication 50 pounds of body weight x ounces of medication 175pounds of body weight 2 ounces x ounces 50 pounds 175 pounds 2 x Cross-multiply: 50( x ) 2(175) Solve for x: 50x x x 7 6. A person who weighs 175 pounds requires 7 ounces of the medication. Exercises: Set up a proportion to solve each of the following word problems. 1. A car averages 12 gallons of gasoline for each 420 miles driven. How much gasoline will it need for a 3,360-mile trip? 2. A $9 sales tax is charged for a $200 purchase. At this rate, what is the sales tax for a $300 purchase? 3. A life insurance policy costs $6.88 for every $1,000 of insurance. At this rate, what is the cost for $17,500 worth of life insurance? 4. A cabinet maker used 3 4 gallons of varnish on 10 cabinet doors. At this rate, how many quarts of varnish are required for 25 cabinets? Revised 05/10 6

7 5. A stock investment of 50 shares paid a dividend of $125. At this rate, what dividend would be paid for 325 shares of stock? 6. A biologist estimates that the human body contains 90 pounds of water for every 100 pounds of body weight. At this rate, estimate the number of pounds of water in a child who weighs 80 pounds. 7. If 21 sandwiches are needed at an outing for 14 people, how many sandwiches would be needed for 30 people. 8. If a pole 18 feet high casts a shadow 20 feet long, how long a shadow would a pole 27 feet high cast? 9. A doctor s prescription calls for 1/5 ounce of a particular ingredient for every 20 pounds of body weight. How many ounces of this ingredient would be needed by someone weighing 175 pounds? 10. A farm containing 36.5 acres sells for $5,840. What would a farm of 124 acres sell for at the same rate? Solutions to the odd-numbered exercises and answers to the even-numbered exercises: 12 gallons x gallons The sales tax on a $ miles 3,360 miles 420( x) 12(3,360) purchase is $ x 40, x 40, x gallons of gas will be needed for a 3,360-mile trip. $6.88cost $ x cost quarts of varnish are $1,000insurance $17,500 insurance 1000( x) 6.88(17,500) required for 25 cabinets. 1000x 120,400 (Hint: Change 3/4 gallons to quarts.) 1000x 120, x The cost for $17,500 worth of insurance is $ Revised 05/10 7

8 5. 50 shares 325 shares $125 $ x 6. A child weighing 80 pounds would 50( x) 125(325) contain about 72 pounds of water. 50x 40, x 40, x A $ dividend would be paid for 325 shares of stock. 21 sandwiches x sandwiches The shadow cast by a pole 27 feet high 14 people 30 people 14( x) 21(30) would be 30 feet long. 14x x x sandwiches would be needed for 30 people ounce 5 x ounces 20 pounds 175 pounds 10. A 124-acre farm would sell 20( x) 1 (175) 5 for $19, x 35 20x x 1.75 Someone weighing 175 pounds would need 1.75 ounces of the ingredient in the prescription. Revised 05/10 8